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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2011 South East Mathematical Olympiad
2011 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(4)
4
2
Hide problems
The same angles
Let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
, a line passes through
O
O
O
intersects sides
A
B
,
A
C
AB,AC
A
B
,
A
C
at points
M
,
N
M,N
M
,
N
,
E
E
E
is the midpoint of
M
C
MC
MC
,
F
F
F
is the midpoint of
N
B
NB
NB
, prove that :
∠
F
O
E
=
∠
B
A
C
\angle FOE= \angle BAC
∠
FOE
=
∠
B
A
C
The 12 points on the clock
12 points are located on a clock with the sme distance , numbers
1
,
2
,
3
,
.
.
.
12
1,2,3 , ... 12
1
,
2
,
3
,
...12
are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct
n
n
n
convex quadrilaterals . They satisfies the following conditions: (1). the colours of vertex of every convex quadrilateral are different from each other . (2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other . Find the maximum
n
n
n
.
3
2
Hide problems
find all n
Find all positive integer
n
n
n
, such that for all 35-element-subsets of
M
=
(
1
,
2
,
3
,
.
.
.
,
50
)
M=(1,2,3,...,50)
M
=
(
1
,
2
,
3
,
...
,
50
)
,there exists at least two different elements
a
,
b
a,b
a
,
b
, satisfing :
a
−
b
=
n
a-b=n
a
−
b
=
n
or
a
+
b
=
n
a+b=n
a
+
b
=
n
.
sequence problem
The sequence
(
a
n
)
n
>
=
1
(a_n)_{n>=1}
(
a
n
)
n
>=
1
satisfies that :
a
1
=
a
2
=
1
a_1=a_2=1
a
1
=
a
2
=
1
a
n
=
7
a
n
−
1
−
a
n
−
2
a_n=7a_{n-1}-a_{n-2}
a
n
=
7
a
n
−
1
−
a
n
−
2
(
n
>
=
3
n>=3
n
>=
3
) , prove that : for all positive integer n , number
a
n
+
2
+
a
n
+
1
a_n+2+a_{n+1}
a
n
+
2
+
a
n
+
1
is a perfect square .
1
2
Hide problems
Function problem
If
min
{
a
x
2
+
b
x
2
+
1
∣
x
∈
R
}
=
3
\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3
min
{
x
2
+
1
a
x
2
+
b
∣
x
∈
R
}
=
3
, then (1) Find the range of
b
b
b
; (2) for every given
b
b
b
, find
a
a
a
.
Collinear problem
In triangle
A
B
C
ABC
A
BC
,
A
A
0
,
B
B
0
,
C
C
0
AA_0,BB_0,CC_0
A
A
0
,
B
B
0
,
C
C
0
are the angle bisectors ,
A
0
,
B
0
,
C
0
A_0,B_0,C_0
A
0
,
B
0
,
C
0
are on sides
B
C
,
C
A
,
A
B
,
BC,CA,AB,
BC
,
C
A
,
A
B
,
draw
A
0
A
1
/
/
B
B
0
,
A
0
A
2
/
/
C
C
0
A_0A_1//BB_0,A_0A_2//CC_0
A
0
A
1
//
B
B
0
,
A
0
A
2
//
C
C
0
,
A
1
A_1
A
1
lies on
A
C
AC
A
C
,
A
2
A_2
A
2
lies on
A
B
AB
A
B
,
A
1
A
2
A_1A_2
A
1
A
2
intersects
B
C
BC
BC
at
A
3
A_3
A
3
.
B
3
B_3
B
3
,
C
3
C_3
C
3
are constructed similarly.Prove that :
A
3
,
B
3
,
C
3
A_3,B_3,C_3
A
3
,
B
3
,
C
3
are collinear.
2
2
Hide problems
a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3)
If positive integers,
a
,
b
,
c
a,b,c
a
,
b
,
c
are pair-wise co-prime, and,
a
2
∣
(
b
3
+
c
3
)
,
b
2
∣
(
a
3
+
c
3
)
,
c
2
∣
(
a
3
+
b
3
)
\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3)
a
2
∣
(
b
3
+
c
3
)
,
b
2
∣
(
a
3
+
c
3
)
,
c
2
∣
(
a
3
+
b
3
)
find
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
Geometric inequality
Let
P
i
P_i
P
i
i
=
1
,
2
,
.
.
.
.
.
.
n
i=1,2,......n
i
=
1
,
2
,
......
n
be
n
n
n
points on the plane ,
M
M
M
is a point on segment
A
B
AB
A
B
in the same plane , prove :
∑
i
=
1
n
∣
P
i
M
∣
≤
max
(
∑
i
=
1
n
∣
P
i
A
∣
,
∑
i
=
1
n
∣
P
i
B
∣
)
\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )
∑
i
=
1
n
∣
P
i
M
∣
≤
max
(
∑
i
=
1
n
∣
P
i
A
∣
,
∑
i
=
1
n
∣
P
i
B
∣
)
. (Here
∣
A
B
∣
|AB|
∣
A
B
∣
means the length of segment
A
B
AB
A
B
) .